120 ideas
19693 | There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb] |
24047 | An account is either a definition or a demonstration [Aristotle] |
1575 | For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle] |
1589 | Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle] |
24052 | From one thing alone we can infer its contrary [Aristotle] |
8200 | Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine] |
4385 | Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson] |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
13282 | Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
4730 | For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady] |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
1729 | We perceive number by the denial of continuity [Aristotle] |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
24057 | What is prior is always potentially present in what is next in order [Aristotle] |
16752 | Sight is the essence of the eye, fitting its definition; the eye itself is just the matter [Aristotle] |
24058 | The substance is the cause of a thing's being [Aristotle] |
24055 | Matter is potential, form is actual [Aristotle] |
13276 | The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki] |
24040 | Scientists explain anger by the matter, dialecticians by the form and the account [Aristotle] |
13277 | The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki] |
5991 | For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code] |
11239 | The notion of a priori truth is absent in Aristotle [Aristotle, by Politis] |
5051 | The intellect has potential to think, like a tablet on which nothing has yet been written [Aristotle] |
1732 | Sense organs aren't the end of sensation, or they would know what does the sensing [Aristotle] |
17711 | Our minds take on the form of what is being perceived [Aristotle, by Mares] |
1725 | Why can't we sense the senses? And why do senses need stimuli? [Aristotle] |
1724 | Perception necessitates pleasure and pain, which necessitates appetite [Aristotle] |
1730 | Why do we have many senses, and not just one? [Aristotle] |
16723 | Perception of sensible objects is virtually never wrong [Aristotle] |
1728 | Many objects of sensation are common to all the senses [Aristotle] |
1727 | Some objects of sensation are unique to one sense, where deception is impossible [Aristotle] |
1734 | In moral thought images are essential, to be pursued or avoided [Aristotle] |
1726 | We may think when we wish, but not perceive, because universals are within the mind [Aristotle] |
23312 | Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M] |
16111 | Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML] |
16647 | Demonstration starts from a definition of essence, so we can derive (or conjecture about) the properties [Aristotle] |
24048 | Demonstrations move from starting-points to deduced conclusions [Aristotle] |
16971 | Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik] |
11243 | Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis] |
3320 | Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA] |
12000 | Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung] |
16646 | To understand a triangle summing to two right angles, we need to know the essence of a line [Aristotle] |
1714 | Mind involves movement, perception, incorporeality [Aristotle] |
5507 | Aristotle led to the view that there are several souls, all somewhat physical [Aristotle, by Martin/Barresi] |
24051 | Soul is seen as what moves, or what is least physical, or a combination of elements [Aristotle] |
12086 | Psuché is the form and actuality of a body which potentially has life [Aristotle] |
16754 | The soul is the cause or source of movement, the essence of body, and its end [Aristotle] |
24046 | Understanding is impossible, if it involves the understanding having parts [Aristotle] |
1717 | If the soul is composed of many physical parts, it can't be a true unity [Aristotle] |
24053 | If a soul have parts, what unites them? [Aristotle] |
1721 | What unifies the soul would have to be a super-soul, which seems absurd [Aristotle] |
1735 | In a way the soul is everything which exists, through its perceptions and thoughts [Aristotle] |
24061 | If we divide the mind up according to its capacities, there are a lot of them [Aristotle] |
24062 | Self-moving animals must have desires, and that entails having imagination [Aristotle] |
1710 | Emotion involves the body, thinking uses the mind, imagination hovers between them [Aristotle] |
24039 | All the emotions seem to involve the body, simultaneously with the feeling [Aristotle] |
24056 | The soul (or parts of it) is not separable from the body [Aristotle] |
24050 | If soul is separate from body, why does it die when the body dies? [Aristotle] |
24049 | Thinkers place the soul within the body, but never explain how they are attached [Aristotle] |
1514 | Early thinkers concentrate on the soul but ignore the body, as if it didn't matter what body received the soul [Aristotle] |
2683 | Aristotle has a problem fitting his separate reason into the soul, which is said to be the form of the body [Ackrill on Aristotle] |
1718 | Does the mind think or pity, or does the whole man do these things? [Aristotle] |
13275 | The soul and the body are inseparable, like the imprint in some wax [Aristotle] |
1733 | Thinking is not perceiving, but takes the form of imagination and speculation [Aristotle] |
23307 | Aristotle makes belief a part of reason, but sees desires as separate [Aristotle, by Sorabji] |
23300 | Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
11240 | The notion of analytic truth is absent in Aristotle [Aristotle, by Politis] |
24060 | Self-controlled follow understanding, when it is opposed to desires [Aristotle] |
6559 | Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin] |
4376 | Pleasure and pain are perceptions of things as good or bad [Aristotle] |
11150 | It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle] |
3037 | Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius] |
1740 | Nature does nothing in vain [Aristotle] |
8660 | There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend] |
12058 | Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins] |
24044 | Movement can be intrinsic (like a ship) or relative (like its sailors) [Aristotle] |
24045 | Movement is spatial, alteration, withering or growth [Aristotle] |
1738 | Practical reason is based on desire, so desire must be the ultimate producer of movement [Aristotle] |
1739 | If all movement is either pushing or pulling, there must be a still point in between where it all starts [Aristotle] |
24064 | If something is pushed, it pushes back [Aristotle] |
24063 | What is born has growth, a prime, and a withering away [Aristotle] |
22729 | The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus] |